# Dimension of a Set of Transformations

If we consider the set R of all linear transformations from an p-dimensional vector space Z to Z (T:Z -> Z), what do we know about the dimension of the set R?

In other words, what do we know about any basis for R? What are its properties?

## Answers and Replies

Hint: those linear mappings can be represented by matrices. What's the dimension of this space of matrices?

Hint: those linear mappings can be represented by matrices. What's the dimension of this space of matrices?

Well, we know that each matrix is pxp, because Z is p-dimensional. The dimension of this space of matrices is p-squared (for instance there are 4 basis matrices for the 2x2 case).
But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

Huh, what do you mean?? Every matrix is linear.

Thanks for staying with me!

Huh, what do you mean?? Every matrix is linear.

I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

On another note, is there a linear transformation that maps every vector in a vector space to zero?

I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

It does represent a linear function, namely f(x,y)=(y,0). Every matrix represents a linear function, that's why matrices are so important.

On another note, is there a linear transformation that maps every vector in a vector space to zero?

Yep, that is a linear transformation. It's the analog of the zero matrix.

Thanks for your help!

Just to make sure, the dimension is p-squared, right?

Right!